Number of distinct normals of a parabola passing through the focus of the parabola

Question

Number of distinct normals of a parabola passing through the focus of the parabola (A) 0 (B) 1 (C) 2 (D) 3

Tardigrade Admin · Accepted Answer

Let the parabola be y2=4 a x Equation of any normal is y=-t x+2 a t+a t3 It passes through the focus (a, 0) 0=-a t+2 a t+a t3=a t(1+t2) ⇒ t =0 ∴ There is only 1 real normal.

Q. Number of distinct normals of a parabola passing through the focus of the parabola

0

1

2

3

Let the parabola be $y^{2} = 4 a x$
Equation of any normal is
$y = - t x + 2 a t + a t^{3}$
It passes through the focus $(a, 0)$
$0 = - a t + 2 a t + a t^{3} = a t (1 + t^{2})$
$\Rightarrow t = 0$
$∴$ There is only 1 real normal.

Q. Number of distinct normals of a parabola passing through the focus of the parabola

0

1

2

3

Let the parabola be y2=4ax Equation of any normal is y=−tx+2at+at3 It passes through the focus (a,0) 0=−at+2at+at3=at(1+t2) ⇒t=0 ∴ There is only 1 real normal.

Let the parabola be $y^{2} = 4 a x$
Equation of any normal is
$y = - t x + 2 a t + a t^{3}$
It passes through the focus $(a, 0)$
$0 = - a t + 2 a t + a t^{3} = a t (1 + t^{2})$
$\Rightarrow t = 0$
$∴$ There is only 1 real normal.